John Cremona on Fri, 13 Dec 2024 20:16:29 +0100


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Re: ellperiods

Thank you Bill.  I don't think I was at that talk of yours.

John

On Fri, 13 Dec 2024, 17:40 Bill Allombert, <Bill.Allombert@math.u-bordeaux.fr> wrote:
On Fri, Dec 13, 2024 at 04:00:06PM +0000, John Cremona wrote:
> It is not clear to me what ellperiods(e) returns when e is an ellinit for a
> curve defined over a number field.

ellperiods assume your curve is defined over R or C.

> Also, if I want the period lattice for
> all the embeddings of the fields into R or C, how would I do that?

Just use E.omega.

Otherwise, you would need to apply all the embeddings and then use ellperiods
for each of them.

> I am trying to see how many of the quantities appearing in the BSD
> conjecture can be computed in GP, over a number field.

Assuming the rank and generators are know, everything except the
Tate-Shafarevich group, including the root number.

> I am assuming
> that we know generators somehow (ellgenerators() is not available), and
> something to replace ellL1() so that we can get something like ellbsd().

Just use lfun(E,1)

> I may have missed functions which are already there for curves over number
> fields -- if so, please tell me.

May I suggest this tutorial ?
https://pari.math.u-bordeaux.fr/Events/PARI2018b/talks/elliptic.pdf
especially the last page.

Cheers,
Bill.